C3 chapter 5 - transforming graphs of functions

"In this chapter you will learn how sketch graphs of modulus functions and solve equations involving modulus functions."

Sections

C3 section 5.1

You need to be able to sketch the graph of the modulus function:
$\displaystyle y = \mid f(x) \mid$

You need to know that the modulus of a number n is written as lnl and that this is its value made positive:
$\displaystyle \mid x \mid = x \; \forall x \ge 0$
$\displaystyle \mid x \mid = -x \; \forall x<0$

On some calculators and computer programs the function abs(x) is used for this purpose.

C3 Exercise 5A

• C3 Exercise 5A worked solutions:

• Questions similar to this exercise:

C3 section 5.2

C3 Exercise 5C

• C3 Exercise 5C worked solutions:

C3 section 5.4

Before you start this section:

• you need to be able to sketch the graph of a range of standard functions:

. . . . . . . .	class of graph:	examples:
	small-power polynomial graphs	$y = x^2$
		$y = x^3$
	reciprocal graphs	$y = \dfrac{1}{x}$
	trigonometrical graphs	$y = \sin{x}$
		$y = \cos{x}$
		$y = \tan{x}$
	the modulus function	$y = |x|$

• you need to be familiar with the transformation of graphs from C1 chapter 4

There are four basic transformations to understand:

	transformation	of a polynomial	of a reciprocal	of a trigonometric	of a modulus
1.	the horizontal translation through - a $y = \text { f}(x + a)$
2.	the vertical translation through + a $y = \text { f}(x) + a$
3.	the horizontal stretch with scale factor 1/a $y = \text { f}(ax)$
4.	the vertical stretch with scale factor a $y = a \text {f} (x)$

A particularly tricky topic seems to be when there is a modulus and another function, so for clarity, here are four cases:

	ordering of function and modulus	cubic	reciprocal	trigonometric	modulus
1.	the modulus of a function of x $y = | \text( f}(x) |$ this reflects any parts of the original function that were below the x-axis to above the x-axis.
2.	the function of the modulus of x $y = \text{f}(|x|)$ this takes the right hand portion of the graph and reflects it in the y-axis to get the left hand side.
3.	negative modulus of a function $y = - | \text{ f}(x) |$ this starts with 1: reflect any pars below to above, now reflect everything you've got to below.
4.	the function of the modulus of the negative $y = \text{f}(|-x|)$ start with the x. What happens to it? Negatives are made positive, and vice versa.

You'd be wise to pay particular attention to the case where we apply the modulus to another function already containing a modulus asa these are common on the exam.

C3 Exercise 5C

C3 chapter 5 assessments

A brief (5 to 10 minute) test to check understanding of modulus functions, translation of graphs and solving simple modulus equations: